2025/09/05 by Norrbo, David
#(Primary) 30H10 #30H20 (Secondary) #47B91 #47G10 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2509.05173
Let X be a reflexive Hardy space or weighted Bergman space on the unit disk in the complex plane. For a bounded linear operator S on X, let \textrmwem(S):= sup(fn) \limsupn ‖Sfn‖, that is, the supremum of cluster points of n↦ ‖S fn‖, where (fn) is any unit norm weakly null sequence. This quantity coincides with the essential norm on the reflexive weighted Bergman spaces. For a suitable family \ gt : t∈]0,1[ \ of bounded analytic functions on the unit disk, we characterize when one can exchange \textrmwem(⋅) and integration over t of the multiplication operators Mgt, that is, when \textrmwem( ∫ Mgt dt ) = ∫ \textrmwem( Mgt ) dt ; when the functions gt,t∈]0,1[ can be continuously extended to the unit circle, we obtain a neat function-theoretic characterization.